If we produce a state of inversion, i.e. we have at least as many electrons in a high energy state as in the low energy state to which they fall be radiating recombinations, we will be able to amplify light. | ||||||||

Let's look at a real light amplifier now. The input light with an intensity enters the material, travels through the length of it, getting amplified all the time, and finally exits with
a higher intensity I_{
0}.I | ||||||||

For a quantitative analysis, let's consider a semiconductor which we keep in inversion
conditions extending from to z = 0 along the z = L-axis. We now shine
some light on it a z and with the spectral intensity z = 0.u_{n}(z
= 0) | ||||||||

By definition, the rate for stimulated emission
events, , will increase in R_{se}-direction and so does the spectral intensity of the
radiation, z which we
now write somewhat simplified as u(n , z). u_{n
}(z) | ||||||||

We also assume that the inversion conditions are the same everywhere (i.e.
they do not depend on ), and now define the znet rate of stimulated emissions in short hand:
| ||||||||

| ||||||||

The interesting quantity, if we want to discuss the amplification of light, is , corresponding to the density of photons that, per second, travel along the u_{n}(z)-axis.
If we have any amplification at all, it will increase for increasing z and the rate of increase is somehow
given by an amplification factor z which must be a function of the "strength" of the inversion condition which
in turn is tied to g_{n}. We must expect that amplification is different at different frequencies and
it is thus wise to index R(z) with "gn ". If is constant (which we can expect for constant g_{n}), the spectral intensity
R at some point u_{n
}(z + Dz)
will be given by z + Dz as follows:u_{n} (z) | ||||||||

| ||||||||

This means that the intensity at the end of some length D
of material is given by the intensity available at the entrance plus the part that is generated in the length increment
considered. This part is proportional to the incremental length zD available for amplification, the factor z
, which is defined by
this equation and which will be properly called g_{n}gain coefficient, and the intensity
available. | ||||||||

Making D arbitrarily small, i.e. moving
from zD to zd, yields a simple differential equation:z |
||||||||

| ||||||||

The solution, of course, is | ||||||||

| ||||||||

If we measure the intensity of the light in some conventional units, we have the same relation, of course,
because any measure of intensity at some frequency I
n is always proportional to the number
of photons. The increase in intensity then is | ||||||||

| ||||||||

Formally, this is nothing but Beer's
law of absorption if we introduce a negative absorption coefficient a,
i.e a = – . g_{n} | ||||||||

While this was fairly straightforward, two questions remain: | ||||||||

First, an obvious question: What determines the gain coefficient? | ||||||||

Second, something a bit less obvious. At this point we have made all kinds of assumptions
and approximations, and it is difficult to keep track of what kind of problem we are considering compared to reality. If
you think about this, it all boils down to the following question: Are there other losses
besides fundamental absorption to the photons and to the electrons in the inversion state that we have not yet included?
Because if there are, we will have a harder time to amplify light than we think we have. Our present major goal, the amplification
of light, then would be harder to achieve. | ||||||||

Those are rather difficult questions which we will only consider summarily in this module. There are, however, links to more advanced stuff in what follows. | ||||||||

Gain Coefficient and Transparency Density | ||||||

Looking back at the simple example
used for defining inversion, it is clear that the gain coefficient
increases if the degree of inversion, i.e. the ratio of stimulated emission to fundamental absorption events, increases.g_{n} |
||||||

This implies that increases
with increasing carrier density in the conduction band which in turn demands that g_{n}, the quasi Fermi energy of the electrons in the conduction band, moves deeper into the conduction band.E_{F}
^{e} |
||||||

The gain coefficient , moreover, will
be largest at the frequency corresponding to the energy levels where most electrons can be found. This level moves up in
energy with increasing density of the electrons; for very few electrons it is of course g_{n}.E_{C} |
||||||

Again, from the simple example used before, we can conclude that for the onset of inversion, i.e., for identical rates of fundamental absorption and stimulated emission, nothing happens in total: Exactly the same number of photons emerges at the output as fed into the input. We have | ||||||

| ||||||

which demands exp ( or gn · z) = 1.g_{n} = 0 | ||||||

The effect now will be that the semiconductor appears completely
transparent to the light. The necessary density of electrons (for some fixed density of holes) is called transparency density
.
n^{e}_{T} | ||||||

If the carrier density increases beyond n^{e},
the maximum value of n^{e}_{T} obtained at a certain frequency (which increases
sightly with g_{n}) will increase, too, in a pretty much linear fashion. We find more or less empirically:n^{e} |
||||||

| ||||||

The factor may simply be considered to be a material constant called adifferential gain factor, since it depends more or less on material parameters like density
of states, band gap, etc., and is hard to calculate for real materials. For
example, GaAs has a value of .a
» 2.4 · 10^{–16} cm^{2} | ||||||

In total, we have a complex dependence of
on carrier density and frequency. An advanced module will give more details.g_{n} |
||||||

Additional Losses
| ||

There are two kinds of possible losses that we may have to worry about. | ||

We may loose some photons somehow which then cannot stimulate electrons to emit another photon. | ||

We may loose electrons in some recombination channels; these electrons then aren't available for stimulated emission. | ||

The first kind of loss includes fundamental absorption, but that is already included in the theory so far. Are there other optical losses in the semiconductor? | ||

Well, there are. Besides fundamental absorption, we also have the kind of
absorption that prevents metals from being transparent: Photons are generally absorbed by the free
carriers, i.e., by the electrons in the conduction band. If we increase the carrier density we will increase
this effect. | ||

The second kind of loss includes all electrons that recombine through one of the other channels available: deep levels, direct recombinations, Auger recombination, excitons, .... . | ||

These recombination events not only reduce the number of available electrons, but the ones
disapperaring via radiative recombination produce some light of their own – with the right wave length but with random phases, i.e., not coherent to the light we care for. This light also becomes amplified
and induces a kind of phase noise. | ||

Luckily, all these recombination losses are negligible for real devices. | ||

There might be more loss mechanisms, but we will sweep 'em all under the rug and
simply combine everything there is (or might be) with respect to intrinsic losses (i.e., inside the semiconductor) in an
intrinsic loss coefficient
a._{i} |

The decrease in intensity due to the intrinsic losses then is | ||||||

| ||||||

We can combine gain and losses then to the final equation linking the input to the ouptput: | ||||||

| ||||||

For a constant gain coefficient along the crystal, the total output is then | ||||||

| ||||||

There is an important consequence from this equation: Amplifications demands that
and that requires g_{n} – a_{i} > 0
.g_{ n} > a_{ i} |
||||||

Just achieving inversion (corresponding to g) thus is not good enough. There is a minimum or _{n} = 0threshold
value given by a before light amplification will occur. In other words: The
density of electrons has to be larger than just the transparency density _{i}.n^{e}
_{T} | ||||||

What this means in practice is shown below in a schematic way. | ||||||

Shown are curves for for g_{n}
GaAs in a halfway realistic manner including some numbers. | ||||||

| ||||||

A constant hole density of n,
i.e., _{h} = 1 · 10^{19} cm^{–3} heavily doped p-type GaAs has been used as a reference. The electron
density is raised by injection to four values marked n through _{1}n . _{4} |
||||||

The gain coefficient is given as a
function of the frequency (in terms of energy). A second scale shows the necessary level of the quasi Fermi energy for the
electrons as g_{n}D above the conduction band edge.
E^{e} _{F} | ||||||

For the electron density n we have the onset of inversion. The gain coefficient
is _{1} for exactly one frequency corresponding to the band gap.g = 0 | ||||||

With increasing n, the gain coefficient is > 0
for a portion of the frequency interval, peaking at about the frequency corresponding to - that's where most of the electrons are!E_{g} +
½(DE^{e}_{F}) |
||||||

Only when is above the intrinsic loss
coefficient g_{n}a, which has been drawn in in a halfway realistic manner, some
amplification occurs in the part of the spectrum indicated._{i} | ||||||

More to this in the advanced module "gain coefficient" | ||||||

© H. Föll (Semiconductors - Script)